Macromolecular complexes form the machinery of life, and are relevant to understanding many diseases such as cancer and metabolic disorders. Knowledge of these structures can provide not only the mechanistic descriptions for how these complexes function but also clues in developing therapeutic interventions related to disease. Prediction of these multi-component biomolecularcomplexes ("protein-protein docking") is a key technology in developing such knowledge. We propose a unique collaborationbetween structural molecular biologists, applied mathematicians and computer scientists to develop and optimize novel algorithms and integrate them into a flexible docking-workflow environment. We envision improving the speed, efficiency, generality and flexibility of predicting, visualizing and analyzing protein-protein interactions. The resulting software shall be calibrated, validated and made freely available to the academic community. Our analysis of current approaches indicates that by utilizing and developing state-of-the-art mathematical models and algorithms we can significantly improve the prediction of protein-protein interactions. These improvements address a variety of limitations in current docking approaches, including: 1) Applicabilityto a wide range of biological systems: The proposed space-efficient, multi-resolution, volumetric representationof molecular shape is usable for any protein topology and it significantly increases the size of systems that can be computed; 2) Extensibility: The proposed representation allows us to capture molecular properties using methods similar to our representation of molecular shape and is extensible to flexible proteins; 3)Space and Time Efficiency: Our novel representation lends itself naturally to extremely rapid space efficient search and scoring utilizing a novel adaptive irregularly sampled Fourier calculation;4) Soft Anisotropic DockingAffinity Functions: Our scoring of protein-protein interactions is based on analyticdocking affinity functions defined on molecular interface volumes allowing for both soft docking as well as the ability, for instance, to model water in the interface and other aspects of biological significance.